CHAPTER 2

REVIEW OF LITERATURE

2.1 INTRODUCTION

Stiffened cylindrical shell forms are extensively used as structural

components in naval and offshore industry. Buckling analysis of these shell forms

are very relevant in subsea applications since the hydrostatic pressure induces

compressive stress resultants in shell membrane, An attempt has been made here to

realize the state of art in the analysis and design of cylindrical shells. Literature

describing early classical closed form solutions as well as finite element analysis of

stiffened cylindrical shells are reviewed and presented under subheadings classical

methods, axisymmetric cylindrical shell finite elements, follower force effect and

design aspects.

2.2 CLASSICAL SOLUTIONS

Classical solutions for linear and buckling analysis of unstiffened cylindrical

shells are available through Timoshenko (1961), Flugge (1962), Donnell (1976),

Novozhilov (1959), Kraus (1967) and Brush and Almroth (1975).

2.2.1 Shell Buckling

The buckling pressure of an unstiffened shell with uniform thickness with

simply supported boundary condition is given by von Mises as eqn. 2.].

Pc=C[~E(t~D)] l l(tlDi [(n

2+m2

i -2n2

+1] + 2m4

2 2 J (2.1)~n +m /2-1~ [ 3(l_~2) (n +m )

Where m = nR/Ls

van Mises' expression is still widely used because it has been presented in a

relatively simple form and gives slightly conservative values (Faulkner, 1983).

Windenburg and Trilling (1934) have developed another simplified equation based

13

on von Mises' to predict the collapse pressure under hydrostatic pressure loading and

this is given as eqn.2.2.

2.24E(t/D)5/2

........................ (2.2)

Analytical solutions for buckling analysis of unstiffened cylindrical shells

are giyen by Batdorf (1947) and Nash (1954).

Reis and Walker (1984) have analysed the local buckling strength of .ring

stiffened cylindrical shells under external pressure. The collapse pressure is

calculated by assuming failure to occur when the material reaches a plastic stress

state. Ross (2000) has observed that many vessels buckle at a pressure that are

considerably less than those predicted by elastic theory and introduced a plastic

knockdown factor PKD by which the theoretical elastic instability buckling pressure

is to be divided, to get the predicted buckling pressure. The value of PKD can be

taken from the semi empirical chart developed by Ross.

2.2.2 Shell Yielding

Von Sanden and Gunther (Cormstock, 1988) have developed two equations

to predict the pressure at which yielding of the shell occurs at frame and midbay,

For yielding at frame

2ay (UD)p =

0.5+1.815K«O.85-B)/(1+P))

For yielding at midbay

2cry(t/D)p

1+H «O.85-B)/(1+P))

........................ (2.3)

......................... (2 ..4)

More exact analysis has been made by Salemo and Pulos to include the

effect of axial loading (Jackson, 1992).

14

2.2.3 General Instability

Classical solutions for general instability of ring-stiffened shells under

hydrostatic pressure are given by Kendrick (1953), Bijlaard (1957) and Galletly

(1957). Kendrick has presented a classical variational formulation of the differential

equation of buckling analysis of ring stiffened cylindrical shells. By assuming a half

sine wave between supports as the buckling deformation and proper allowance for

shell distortions between frames, collapse pressure has been predicted by Kendrick

(1965) using Ritz's procedure for simply supported - simply supported boundary

conditions and has been extended for clamped boundary condition by Kaminsky

(1954). Displacement field used by Kendrick has been modified by Ross (1965) and

general instability analysis of ring stiffened cylindrical shells has been performed

incorporating various degrees of rotational restraint at the boundary.

Bresse has developed an expression for elastic collapse of infinitely long

ring-framed compartments (Timoshenko, 1961). Bryant has modified the formula

developed by Kendrick by combining van Mises' and Bresse's relations for the

determination of the overall buckling pressure of ring stiffened cylindrical shell with

simply supported boundary conditions and is available in the form as,

Buckling pressure of stiffened cylindrical shell P, = Per + Pes (2.5)

Pcf = buckling pressure of ring stiffeners = {[n2-1] EI/R3L}

Pcs = buckling pressure of shell = Et/R {m4/([n2_1+(m2/2)][n2+m2] 2)} (2.6)

and m = 1tR/Ls

Bryant's two-term approximation to the overall buckling pressure has

gained wide acceptance because of its simplicity (Faulkner, 1983). The effect of

imperfections on buckling pressure has been investigated and an expression has been

developed by Bijlaard (1957).

The critical pressure for general instability of ring stiffened, stringer

stiffened and ring and stringer stiffened cylindrical shells are computed by Bodner

(1957). Baruch and Singer (1963) have carried out general instability analysis of

stiffened cylindrical shell by considering the distributed eccentric ring stiffeners and

stringers separately. The well-known superiority of rings over stringers for

15

cylindrical shells under external pressure is very clearly brought out. The effect of

eccentricity of stiffeners is more pronounced for rings than for stringers. Voce (1969)

has developed a solution procedure based on energy method for general instability of

orthotropic ring stiffened cylinders under external hydrostatic pressure for simply

supported boundary condition. Kempner et al (1970) have developed a procedure to

determine the stresses and deflections incorporating the effects of large rotations,

initial deflections and thick shell effects. Singer (1982) has extended buckling

analysis for imperfect stiffened shells. Wu and Zhang (1991) have developed a

nonlinear theoretical analysis for predicting the buckling and post buckling loads of

discretely stiffened cylindrical shells.

Karabalis (1992) has made a simplified analytical procedure, which can be

used as an effective method in checking the design of stiffening frames of cylindrical

fuselages with or without cutouts for failure by general instability. The general

instability mode of failure of cylindrical shell is independent of geometric

discontinuity like cutouts. Any loss in moment of inertia due to the cutouts must be

proportionately compensated by gain in bending stiffness, which can be realized by

the addition of reinforcement possibly at the edges of the cutouts. However large

reinforced cutouts would fail due to local instability at the edges of the cutouts. It is

recommended that the proposed criteria can be used for design and calculation in the

absence rigorous finite element analysis. Huang and Wierzbicki (1993) have

developed a simple analytical model that describes the plastic behaviour of a curved

cylindrical panel with ring stiffeners. Energy methods are used to analyse the plastic

tripping response of the structure. In order to derive a closed form solution to the

problem, a number of simplifications are made such as the material is treated as fully

plastic and the energy corresponding to lateral bending of stiffeners are neglected.

Tian et al (1999) have carried out elastic buckl ing analysis of ring stiffened

cylindrical shells using Ritz's procedure, which can be used as a reference source for

checking the validity of other numerical methods and software for buckling of

cylindrical shells.

Barlag and Rothert (2002) have developed an idealization concept for

stability analysis of ring reinforced cylindrical shells under external pressure..A

16

monograph is introduced based on the stability equation to determine the local and

global buckling pressures of ring stiffened cylindrical shells under external pressure

based on Flugge's strain displacement relations.

The scope of the classical methods is limited to simple boundary

conditions, uniform shell thickness, regular stiffeners and uniform spacing.

2.3 AXISYMMETRIC CYLINDRICAL SHELL FINITE ELEMENTS

Axisymmetric cylindrical shell elements are singly curved, straight meridian

elements. A few relevant papers on axisymmetric shell elements have been reviewed

and presented. Review of literature on finite element modeling of unstiffened and

stiffened cylindrical shells is described subsequently.

2.3.1 Unstiffened Shells

Grafton and Strome (1963) have presented the conical segment elements for

the analysis of shells of revolution. Improvements in the derivation of element

stiffness matrix are presented by Popov et al (1964). Percy et al (1965) have

extended these formulations for orthotropic and laminated materials.

Navaratna et al (1968) have made a linear bifurcation buckling analysis of

unstiffened shells using an axisymmetric rotational finite element in which the

membrane displacements are approximated by linear polynomials and the radial

displacement by cubic polynomial. Trigonometric functions are used to characterize

the buckling waves in circumferential direction. Later this element has been used to

study the influence of out of roundness on buckling theory of unstiffened shells. A

systematic procedure to obtain the geometric stiffness matrix and subsequently the

buckling load through variational approach is presented. Me Donald and White

(1973) have studied the effect of out of roundness in buckling strength of unstiffened

shells. Ross (1974) has carried out lobar bifurcation buckling analysis of thin walled

cylindrical shells under external pressure using axisymmetric finite element based

linear-linear-cubic shape functions. Venkiteswara Rao et al (1974) have reported a

rigorous linear buckling analysis using axisymmetric finite element based all-cubic

shape functions. Surana (1982) has developed a nonlinear formulation for

17

axisymmetric shell elements. Cook has (1982) developed a finite element model for

nonlinear analysis of shell of revolution. Rajagopalan and Ganapathy Chettiar (1983)

have developed an all-cubic axisymmetric rotational shell element for modeling the

cylindrical shell in the interstiffener buckling analysis. Ross and Mackeny (1983)

have carried out deformation and stability studies ofaxisymmetric shells under

external hydrostatic pressure using linear-linear-cubic axisymmetric finite elements.

Gould (1985) has formulated and used axisymmetric shell elements for linear and

nonlinear analysis.

Rajagopalan (1993) has developed a reduced cubic element based on

condensation concept for stability problems. Internal nodes are introduced in the

axisymmetric cylindrical shell element so as to permit cubic polynomial to be taken for

modeling the membrane displacement in the meridional direction. The internal nodes

are eliminated by geometric condensation procedure so that the condensed element

will have only fewer degrees of freedom and hence computationally efficient.

Ross et al (1994) have carried out vibration analysis ofaxisymmetric shells

under external hydrostatic pressure. Both shell and surrounding fluid are discretized

as finite elements. It is reported that dynamic buckling can take place at a pressure

less than that of static buckling pressure.

Koiter et al (1994) have investigated the influence ofaxisymmetric

thickness variation on the buckling load of an axially compressed shell. Mutoh et al

(1996) have presented an alternate lower bound analysis to elastic buckling collapse

of thin shells of revolution. Axisymmetric rotational shell elements whose strain

displacement relations are described by Koiter's small finite deflection theory have

been used for the analysis. In this element the displacements are expanded

circumferentially using a Fourier series.

Sridharan and Kasagi (1997) have presented a summary of the work carried

out in Washington University on buckling and associated non-linear responds and

collapse of moderately thick composite cylindrical shells.

18

Ross et al (2000) have carried out the inelastic buckling analysis of circular

cylinders of varying thickness under external hydrostatic pressure. Analytical results

are verified by experimental investigations. Gusic et al (2000) have analysed the

influence of circumferential thickness variation on the buckling of cylindrical shells

under external pressure by means of finite element bifurcation analysis. Two

different finite element codes, one with quasi-axisymmetrical multimode Fourier

analysis and the other with 3D shell element are used. Numerical integration of

Fourier series permits the introduction of geometric and thickness imperfections at

the integration points.

Correia et al (2000) have used higher order displacement fields with

longitudinal and circumferential components of displacements as power series and

the condition of zero stress at top and bottom surfaces of the shell are imposed.

Combescure and Gusic (2001) have carried out nonlinear buckling analysis ofcylinders under external pressure with nonaxisymmetric thickness imperfections

using axisymmetric shell elements. Gould and Hara (2002) have reported recent

advances in the finite element analysis of shell of revolution. Sze et al (2004) have

discussed about popular benchmark problems for geometric nonlinear shell analysis.

2.3.2 Stiffened Shells

Ross (1976) has carried out stability analysis of ring reinforced circular

cylindrical shells under external hydrostatic pressure. Subbiah and Natarajan (1981)

have carried out a finite element analysis for general instability of ring-stiffened

shells of revolution using axisymmetric shell elements. They have used linear-linear­

cubic element for the finite element modeling of the shells. This smeared model

analysis predicted a lower bound buckling pressure. Influence of various boundary

conditions on buckling pressure has been investigated and reported. A rigorous

derivation for potential due to hydrostatic loading as follower force and subsequent

reduction in buckling pressure has been reported.

Subbiah (1988) has made a nonlinear analysis of geometrically imperfect

stiffened shells of revolution. A nonlinear large deformation finite element analysis

has been carried out for the general instability of ring stiffened cylindrical shells

19

subjected to end compression and circumferential pressure. Smeared model

technique is adopted. A combined nonlinear and eigen value analysis is presented to

determine the critical pressure for initially imperfect stiffened cylinders. The

buckling pressures of thin shell structures are very much sensitive to initial

imperfections. This is one of the major reasons for poor correlation between

theoretically predicted and experimentally obtained buckling loads. The only way to

overcome this discrepancy is to analyse the shell as a nonlinear large deformation

problem with initial imperfections.

Rajagopalan (1993) has used a discrete ring stiffener element and

axisymmetric cylindrical shell element to model the stiffened cylindrical shell.

General buckling analysis has been carried out by rigorous stiffener modeling using

annular plate bending elements and shell elements. The superelement modeling of

stiffeners introduces off shell nodes, which are eliminated by geometric condensation

procedure. Ross (1995) has carried out plastic buckling analysis of ring stiffened

cylindrical shells under external hydrostatic pressure.

Kasagi and Sridharan (1995) have investigated the imperfection sensitivity

of ring stiffened anisotropic composite cylindrical shells under hydrostatic pressure

using an asymptotic procedure. The displacement function takes the form of exact

trigonometric function along the circumferential direction and p-version in other two

directions. Sridharan (1995) has extended an analysis of stiffened cylindrical shells

under interactive buckling. Effects of interaction of local and overall buckling is

analysed using finite elements, in which the local buckling information is embedded.

Schokker et al (1996) have carried out dynamic instability analysis of ring stiffened

composite shells under hydrostatic pressure.

Stanley and Ganesan (1997) have investigated the natural frequencies of

stiffened cylindrical shell (both short and long) with clamped boundary condition.

Two nodded cylindrical shell element with four degrees of freedom per node is used.

20

2.4 RING STIFFENED CYLINDRICAL SHELLS WITH OTHER TYPES OF

FINITE ELEMENTS

Kohnke et al (1972) have made a finite element analysis for eccentrically

stiffened cylindrical shells using 48 degree of freedom shell elements. Giacofci

(1981) has developed modeling techniques for the analysis of stiffened shell

structures. Tsang and Harding (1987) have made plastic and elastic analysis of ring

stiffened cylindrical shells by using finite element program FINAS. Zhen and Yeh

(1990) have developed a new method of analysis capable of predicting nonlinear

buckling load for stiffened cylindrical shells. Pegg (1992) has made a numerical

study of dynamic buckling of ring-stiffened cylinders using general shell elements.

Omurtag and Akoz (1993) have developed mixed finite element formulation for

eccentrically stiffened cylindrical shells. A rectangular four nodded shell element and

a two nodded circular bar element are used for the analysis. Chen et al (1994) have

carried out buckling analysis of ring stiffened cylindrical shells with cutouts by

mixed method of finite strip and finite elements. Finite strip and finite elements are

connected together by specially developed transition elements. Goswami and

Mukopadhyay (1995) have carried out geometrically nonlinear analysis of laminated

stiffened shells. Li et al (1997) have made an adaptive finite element analysis method

for shells with stiffeners.

2.5FOLLOWER FORCE EFFECT

Bodner (1958) has described the buckling of infinitely long cylindrical shell

under various distributed load systems with and without considering the follower

force effect. The buckling load for hydrostatic pressure is found to be lower than that

for the uniformly distributed conservative load system.

Hernnan and Bungay (1964) have studied the stability of elastic system

subjected to nonconservative forces. Oden (1970) has developed an approximate

method for computing nonconservative generalized forces on large deformation

problems.

21

Hibbit (1979) has discussed about the importance of coupling of the

follower force effect with the tangent stiffness matrix of the structure for the accurate

solution of the problems. In presence of free loaded ends, the system become

nonconservative, hence leads to an unsymmetric matrix .. Loganathan et al (1979)

have carried out a study of effect of pressure stiffness in shell stability analysis. The

analysis is carried out in deep and shallow shell situations with and without pressure

stiffness matrix. The analysis without follower force effect leads to bifurcation

buckling modes and with pressure stiffness matrix, the mode of instability changes to

a limit point phenomenon. In general, the inclusion of pressure rotation effect will

introduce unsymmetric stiffness matrices into the finite element equations. Under

such circumstances, the classical bifurcation concept is no longer valid. The solution

of unsymmetric simultaneous system of algebraic equations is very tedious. But in

some cases, such as uniform external pressure on cylindrical shells, the pressure

stiffness matrix is symmetric. Although the problem of follower forces is in general

a noncoservative-loading problem, the symmetric matrix is conservative in character.

Mang (1980) has derived techniques to impose symmetricability to pressure stiffness

matrix. According to him the buckling pressure derived for a cylindrical shell with

unsymmetric pressure stiffness matrix differs very little from the buckling pressure,

resulting from an alternative symmetric pressure stiffness matrix.

Subbiah and Natarajan (1981) have analysed the follower force effect of

hydrostatic pressure in the finite element analysis for general instability of ring­

stiffened shells of revolution using axisymmetric shell elements. A rigorous

derivation for potential due to hydrostatic loading including follower force effect has

been presented. Substantial reduction in buckling pressure due to follower force

effect has been reported. Carnoy et al (1984) have carried out static buckling analysis

of shells subjected to follower pressure by finite element method. Tomski and

Przybyski (1987) have studied the behaviour of a clamped, elastically supported

planar structure under follower force.

Hasegawa et al (1988) have investigated the elastic instability and nonlinear

finite displacement behaviour of special thin walled members under displacement

dependant loadings. When the load stiffness matrix is un symmetric indicating the

nonconservativeness of the load, the dynamic stability becomes a matter of great

22

concern and hence the mass matrices of the special thin walled members are derived

in the study to examine its possibility. The methods of analysis presented in the paper

have been of four types, static instability analysis called divergence, dynamic

instability analysis called flutter, static nonlinear finite displacement analysis and

static linearised finite displacement analysis.

2.6 DESIGN ASPECTS OF SUBMARINE HULLS

Faulkner (1983) has made a discussion about the design practices used in

BS 5500 (1976). According to him the interframe shell collapse determines the main

weight and cost and safety factors should be chosen by ensuring this as the prime

mode of failure. This paper is not meant to provide a comprehensive coverage of

structural design but concentrated on the philosophy and underlying essentials of

strength formulations and design.

Gorman and Louie (1991) have developed an optimization methodology,

which explicitly considers shell yielding, lobar buckling, general instability and local

frame instability failure modes. Quantitative results on the effects of hull circularity

is also presented. Some novel results for the buckling performance of

nonaxisymmetric rings are further presented to identify the design payoff of new

software tools. Empirical relations are used to get the principal characteristics desired

of pressure hull material from weight displacement ratio. The hull wall architecture

has also been commented.

Jackson (1992) describes the concepts of design that has been developed

over a number of years. The optimum length to diameter ratio is 4 to 6. Neto et ,al

(1996) have determined the collapse pressures of ring stiffened cylindrical sheIls

under hydrostatic pressure using code formulations and elastic plastic finite element

analysis.

Bushnell and Bushnell (1996) have developed an approximate method for

the optimum design of ring and stringer stiffened cylindrical shell panels and shells

with imperfections. The PANDA.2 computer program for minimum weight design

of stiffened composite panel is expanded to handle optimization of ring and stringer

stiffened cylindrical panels and shells with three types of initial imperfections in

23

the form of buckling modes, any combination of which may be present; local, inter­

ring and general.

Das et al (1997) have made a reliability based design procedure of stiffened

cylinder using multiple criteria of optimization techniques. The various limit states of

orthogonaly stiffened cylindrical shells have been used and they include bay

instability, frame bending and frame tripping. A rational comprehensive analysis is

required for a safe effective design.

2.7 SCOPE AND OBJECTIVES

For the linear analysis of ring stiffened cylindrical shell with simple

boundary conditions, closed form solutions are available. However, a definite

necessity is felt for the solution of the problem for various practical configurations

and boundary conditions. Finite element method can be adopted for the analysis of

stiffened cylindrical shells owing to its versatility. Finite element modeling of

stiffened cylindrical shell can be done either using a stiffener shell model or a

smeared model. The hydrostatic pressure acting at a considerable depth can be

treated as uniformly distributed pressure loading and consistent load vector can be

formulated. Efficient cylindrical shell elements and circular stiffener elements are

available in the literature, which can be employed for the analysis of subsea stiffened

cylindrical shells. The analytical investigations of cylindrical shells constituting the

submarine hull are classified documents and are rarely found in literature; hence it is

found apt to carryout such investigations to provide design recommendations. 'A

definite need is felt to have a software based on an efficient finite element to analyse

the stiffened cylindrical shell for various boundary conditions, incorporating the

follower force effect.

Scope of the work is to conduct linear elastic, linear buckling and geometric

nonlinear analysis of stiffened cylindrical submarine shells incorporating the

follower force effect of hydrostatic pressure.

24

Theobjectives of the thesis are listed below.

• To develop a software based on all-cubic axisymmetric cylindrical shell finite

element and discrete ring stiffener element for linear elastic, linear buckling

and geometric nonlinear analysis of stiffened cylindrical shells.

• To implement the software in pc environment and use it to predict the stress

resultants, linear buckling pressures and collapse pressures for various

boundary conditions and configurations of the shell and stiffener.

• To study the influence of follower force effect due to hydrostatic pressure on

the collapse pressure of stiffened cylindrical submarine shells.

25